Spatially resolved electromagnetic property measurement

ABSTRACT

A scanning probe detects phase changes of a cantilevered tip proximate to a sample, the oscillations of the cantilevered tip are induced by a lateral bias applied to the sample to quantify the local impedance of the interface normal to the surface of the sample. An ac voltage having a frequency is applied to the sample. The sample is placed at a fixed distance from the cantilevered tip and a phase angle of the cantilevered tip is measured. The position of the cantilevered tip is changed relative to the sample and another phase angle is measured. A phase shift of the deflection of the cantilevered tip is determined based on the phase angles. The impedance of the grain boundary, specifically interface capacitance and resistance, is calculated based on the phase shift and the frequency of the ac voltage. Magnetic properties are measured by applying a dc bias to the tip that cancels electrostatic forces, thereby providing direct measurement of magnetic forces.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional PatentApplication Ser. No. 60/262,347 entitled “Scanning Impedance MicroscopyOf Interfaces,” filed Jan. 18, 2001, which is hereby incorporated byreference in its entirety.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

The invention was supported in part by funds from the U.S. Government(MRSEC Grant No. NSF DMR 00-79909) and the U.S. Government may thereforehave certain rights in the invention.

FIELD OF THE INVENTION

The invention relates generally to electromagnetic property measurement,and more particularly, to spatially resolved impedance microscopy ofinterfaces and to distinguishing surface potential induced forces frommagnetically induced forces for current carrying materials.

BACKGROUND OF THE INVENTION

Many classes of devices function based on the structure, topology, andother properties of grain boundaries or interfaces. Examples includevaristors, PTCR thermistors, diodes, chemical sensors, and solar cells.The properties of interfaces have been extensively studied bymacroscopic techniques, such as dc transport measurements and impedancespectroscopy, etc. These techniques address averaged properties ofinterfaces and little or no information is obtained about the propertiesof an individual interface. Recently, a number of approaches have beensuggested to isolate individual grain boundaries using micropatternedcontacts or bicrystal samples. A number of works accessingcurrent-voltage (I-V) characteristic of single interfaces also have beenreported; however, a major limitation of such techniques is a presetcontact pattern, which does not yield spatially resolved information.Moreover, contact resistance and contact capacitance are included in themeasurements, which may decrease accuracy and complicate datainterpretation. Scanning probe microscopy (SPM) techniques have beensuccessfully used to detect stray fields over Schottky double barriersand to image potential drops at laterally biased grain boundaries;however, the information provided by SPM has been limited to static ordc transport properties of grain boundaries.

SPM techniques based on the detection of tip-surface capacitance and dcresistivity are well known, and some are capable of detecting thefrequency dependence of tip-surface impedance. However, such techniquesdo not quantify the local impedance of an interface normal to thesurface, i.e., local characterization of ac transport properties of aninterface.

Further, some techniques provide force gradient images of interfaces,such as conventional magnetic-force microscopy (MFM). MFM is a dual passtechnique based on detecting the dynamic response of a mechanicallydriven cantilever a magnetic field. During a first pass, theferromagnetic tip of the cantilever acquires a surface topology profileof a sample in an intermittent contact mode. Then, during a second pass,the cantilever is driven mechanically and the surface topographicprofile is retraced at a predefined tip-to-sample surface separation.The magnetic force F_(magn) between the tip and the sample surfacevaries along the length of the sample, thereby causing a change incantilever resonant frequency that is proportional to the force gradientand is given by: $\begin{matrix}{{\Delta\omega} = {\frac{\omega_{o}}{2k}\frac{\mathbb{d}{{Fmagn}(z)}}{\mathbb{d}z}}} & \text{Equation~~1}\end{matrix}$where k is the cantilever spring constant, ω₀ is the resonant frequencyof the cantilever, and z is the tip to surface separation distance.Resonant frequency shift, Δω, data is collected and arranged as a MFMimage of the sample. Image quantification in terms of surface and tipproperties is complex due to the non-local character of the tip-surfaceinteractions. In one point probe approximation, the magnetic state ofthe tip is described by its effective first and second order multipolemoments. The force acting on the probe is given by:F=u ₀(q+m∇)H  Equation 2where q and m are effective probe monopole and dipole moments,respectively, H is the stray field above the surface of the sample, andu₀ is the magnetic permeability of a vacuum (1.256×10⁻⁶H/m). Theeffective monopole and dipole moments of the tip can be obtained bycalibrating against a standard system, for example, micro-fabricatedcoils or lines carrying a known current. These can be used to quantifythe MFM data. However, the total force gradient over the sample may beaffected by electrostatic interactions.

Therefore, a system and method that quantifies the local impedance ofthe interface normal to the sample surface and that overcomes errorsintroduced by electrostatic interactions would be desirable.

SUMMARY OF THE INVENTION

An aspect of the invention is directed to a scanning probe techniquebased on detecting phase change and amplitude of a cantilevered tipproximate to a sample surface, where the oscillations of thecantilevered tip are induced by a bias laterally applied to the sample,and the local impedance of the interface normal to the surface of thesample is quantified.

Frequency dependence of local phase angle and oscillation amplitude canbe used to determine resistance and capacitance of the interface, givena known current limiting resistor. The frequency dependent interfaceimpedance in the parallel R-C approximation is defined by capacitance,C, and resistance, R, as Z=1/(1/R+iωC). Variation of dc bias offsetacross the interface can be used to determine capacitance—voltage (C-V)and resistance—voltage (R-V) characteristics of the interface, whereboth capacitance across a grain boundary, C_(gb), and resistance acrossa grain boundary, R_(gb), can be voltage and frequency dependent.

According to another aspect of the invention, magnetic properties of asample are determined by applying an ac bias to the tip, wherein the acsignal is set to the resonant frequency of the cantilevered tip. A dcbias is applied to the tip and adjusted to cancel the surface potentialof the sample. An ac bias is applied to the sample and magnetic fieldsare determined. The determined magnetic fields do not substantiallyinclude interactions from electrostatic forces.

These features, as well as other features, of the invention will bedescribed in more detail hereinafter.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention is further described in the detailed description thatfollows, by reference to the noted drawings by way of non-limitingillustrative embodiments of the invention, in which like referencenumerals represent similar parts throughout the several views of thedrawings, and wherein:

FIG. 1 is a block diagram of a measurement system in accordance with theinvention;

FIG. 2 is a graph showing measured phase and amplitude signals versusdistance across a grain boundary region obtained from the system of FIG.1;

FIGS. 3 a and 3 b are graphs showing measured cantilever oscillationamplitude versus tip bias and driving voltage, respectively, obtainedfrom the system of FIG. 1;

FIGS. 3 c and 3 d are graphs showing average measured phase versus tipbias and driving voltage, respectively, obtained from the system of FIG.1;

FIGS. 3 e and 3 f are graphs showing grain boundary phase shift versustip bias and driving voltage, respectively, obtained from the system ofFIG. 1;

FIGS. 4 a and 4 b are graphs showing average phase shift and grainboundary phase shift, respectively, versus driving frequency, obtainedfrom the system of FIG. 1;

FIG. 5 a is a circuit diagram of an equivalent circuit for surfacescanning potential microscopy (SSPM) analysis, which can be used inembodiments of the invention to obtain resistance and capacitance valuesof an interface;

FIG. 5 b is a graph showing grain boundary potential drop as a functionof bias, obtained from the system of FIG. 1;

FIG. 6 is a graph showing potential drop at the interface versus lateralbias (i.e., sample bias) for various values of R (current limitingresistor), obtained from the system of FIG. 1;

FIGS. 7 a and 7 b are graphs showing phase shift versus distance alongthe sample path for various values of R (current limiting resistor),obtained from the system of FIG. 1;

FIGS. 8 a and 8 b are graphs showing phase and amplitude, respectivelyversus frequency, obtained from the system of FIG. 1;

FIGS. 9 a and 9 b are graphs showing tan (phase shift) and amplituderatio, respectively versus frequency, obtained from the system of FIG.1;

FIG. 10 a is a graph showing tan (phase shift) versus lateral bias,obtained from the system of FIG. 1;

FIG. 10 b is a graph showing 1/capacitance² versus potential drop,obtained from the system of FIG. 1;

FIG. 11 is an optical micrograph of an exemplary sample that wasanalyzed using the invention;

FIGS. 12 a, 12 c, and 12 e are graphs illustrating calculated forceprofiles and FIGS. 12 b, 12 d, and 12 f are graphs illustrating measuredforce profiles using basic potential correction magnetic-forcemicroscopy (PMFM) mode, obtained from the system of FIG. 1;

FIG. 13 a is a graph of force versus distance illustrating a forceprofile, obtained from the system of FIG. 1;

FIG. 13 b is a graph of tip bias versus measured force illustrating tipbias dependence, obtained from the system of FIG. 1;

FIG. 13 c is a graph of line bias versus measured force at a nulling tipbias, obtained from the system of FIG. 1;

FIG. 13 d is a graph of frequency versus measured force, obtained fromthe system of FIG. 1;

FIG. 14 a is a graph of lateral sample distance versus determined forceat the nulling tip bias, obtained from the system of FIG. 1;

FIG. 14 b is a graph of lateral sample distance versus determinedpotential, obtained from the system of FIG. 1;

FIG. 14 c is a graph of lateral sample distance versus a difference ofthe force and potential profiles of FIGS. 14 a, 14 b, obtained from thesystem of FIG. 1; and

FIG. 14 d is a graph of lateral sample distance versus force ofexperimental data measured while adjusting the dc bias of the tip tomatch surface potential, obtained from the system of FIG. 1.

DETAILED DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS

FIG. 1 illustrates a block diagram of a measurement system in accordancewith an embodiment of the invention. As shown in FIG. 1, the localimpedance of a sample 10 is determined across a grain boundary 11. Abias is laterally applied to the sample 10, across the grain boundary11, and the deflection of a cantilevered tip 15 is measured. Tip 15 maybe magnetic or non-magnetic. For determining grain boundary impedance,tip 15 is typically non-magnetic. The sample bias is applied with afunction generator 20. The magnitude and phase of the deflection ofcantilevered tip 15 are measured with a conventional atomic-forcemicroscope (AFM) controller 25 (Nanoscope-III, Digital Instruments,Santa Barbara, Calif.). Lock in amplifier 27 is coupled to functiongenerator 20 and AFM controller 25. AFM controller 25 may include aprocessor for processing measurements taken from tip 15; alternatively aseparate processor may be coupled to AFM controller 15.

Tip 15 traverses the tip along a predetermined path on the surface ofthe sample while tip 15 is disposed proximate the surface of the sample.During the traversing, the sample surface topography along the path isdetermined. An ac voltage is applied to the sample, laterally across theinterface. With the ac voltage applied to the sample, tip 15 retracesthe path at a predetermined tip-to-sample distance. The response of tip15 is measured and analyzed to produce information about the impedanceof the interface. The measurements include phase and amplitudemeasurements of the cantilevered tip 15 and can yield an impedanceproduct or can be combined with other known techniques to yield aresistance and a capacitance of the interface, thereby providingspatially resolved impedance information.

The system of FIG. 1 was used to take measurements of a Σ5 grainboundary in a Nb-doped SrTiO₃ bicrystal. Tip 15 first traces and pathalong the sample, across the interface and acquires surface topographyin intermittent contact mode and then retraces the surface profilemaintaining constant tip-surface separation, i.e., a first pass and asecond pass. In the first pass, the tip was static (i.e., neithermechanical nor voltage modulation is applied to the tip). In the secondpass, lock-in amplifier 27 (Model SRS830, Stanford Research Systems,Sunnyvale, Calif.) was used to determine the magnitude and phase of thedeflection of the cantilevered tip. The magnitude and phase are relatedto the first harmonic of the force acting on dc biased tip 15. Lock-inamplifier 27 phase offset was set to zero with respect to the functiongenerator 20 output (Model DS340, Stanford Research Systems, Sunnyvale,Calif.). The output amplitude and phase shift, θ, were recorded by theAFM controller 25.

Using this recorded data, amplitude and phase can be graphed versusdistance along the sample path. FIG. 2 shows an exemplary graph ofdeflection amplitude 35 and phase 30 versus distance along the samplepath. As can be seen, the phase shift occurs approximately at a distanceof 7 um from the beginning of measurements, as the phase values shiftfrom positive values to negative values.

Grain boundary position was detected by potential variation due to strayfields of the Double Schottky Barrier (DSB) in a grounded crystal or thepotential drop for a laterally biased crystal by scanning surfacepotential microscopy (SSPM). To perform measurements under bias, SrTiO₃bicrystals were soldered with indium to copper contact pads and anexternal ac or dc bias was supplied by function generator 20.

Two-point dc transport properties of the crystal were independentlyprobed by an HP4145B Semiconductor Parameter Analyzer (Hewlett Packard).AC transport properties were measured by an HP4276A LCZ meter (HewlettPackard) in the frequency range 0.2-20 kHz.

Application of an ac bias having a frequency ω across the grain boundaryresults in a phase shift, φ_(gb), such that: $\begin{matrix}{{\tan\left( \varphi_{gb} \right)} = \frac{\omega\quad C_{gb}R_{gb}^{2}}{\left( {R + R_{gb}} \right) + {R\quad\omega^{2}C_{gb}^{2}R_{gb}^{2}}}} & \text{Equation~~3}\end{matrix}$where C_(gb) and R_(gb) are grain boundary capacitance, grain boundaryresistance, respectively. R is the resistance of a current limitingresistor (circuit termination resistance) in the circuit biasing thesample.In the high frequency limit (defined herein as the region in whichtan(φ_(gb))˜ω⁻¹): $\begin{matrix}{{\tan\left( \varphi_{gb} \right)} = \frac{1}{\omega\quad R\quad C_{gb}}} & \text{Equation~~4}\end{matrix}$In the low frequency limit (defined herein as the region in whichtan(φ_(gb))˜ω): $\begin{matrix}{{\tan\quad\left( \varphi_{gb} \right)} = \frac{\omega\quad C_{gb}R_{gb}^{2}}{\left( {R + R_{gb}} \right)}} & \text{Equation~~5}\end{matrix}$Therefore, analysis in the high frequency limit yields C_(gb) andanalysis in the low frequency limit yields R_(gb) and C_(gb). Crossoverbetween the two limits occurs at a frequency of ω_(r), which is givenby: $\begin{matrix}{\omega_{r}^{2} = \frac{R + R_{gb}}{R\quad C_{gb}^{2}R_{gb}^{2}}} & \text{Equation~~6}\end{matrix}$at which tan(φ_(gb)) has its maximum value of: $\begin{matrix}{{\tan\quad\left( \varphi_{gb} \right)} = \frac{R_{gb}}{2\sqrt{R\left( {R + R_{gb}} \right)}}} & \text{Equation~~7}\end{matrix}$It should be noted that for high R termination, the crossover occurs atω=1/R_(gb)C_(gb), i.e., intrinsic relaxation frequency of the interface,while for low R termination, the resonance frequency shifts to highervalues. However, introduction of high R in the circuit decreases theamplitude of measured signal; therefore, the best results can beobtained when R is close to R_(gb). Alternatively, the value of R can bevaried within several orders of magnitude to obtain quantitative data.In this fashion, the presence of stray resistive (and capacitive)elements can also be detected.

The bias to the sample simultaneously induces an oscillation in surfacepotential, V_(surf), according to:V _(surf) =V _(s) +V _(ac) cos(ωt+φ _(c)),  Equation 8where φ_(c) is the phase shift in the circuit and V_(s) is dc surfacebias. The bias results in a periodic force acting on the dc biased tip.The amplitude, A(ω), and phase, φ, of the cantilever response to theperiodic force F(t)=F_(1ω) cos(ωt) are: $\begin{matrix}{{{A(\omega)} = {\frac{F_{1\omega}}{m}\frac{1}{\sqrt{\left( {\omega^{2} - \omega_{0}^{2}} \right)^{2} + {\omega^{2}\gamma^{2}}}}}}\text{and}} & \text{Equation~~9a} \\{{\tan(\varphi)} = \frac{\omega\quad\gamma}{\omega^{2} - \omega_{0}^{2}}} & \text{Equation~~9b}\end{matrix}$where m is the effective mass, γ is the damping coefficient and ω₀ isthe resonant frequency, of the cantilever. The first harmonic of theforce is: $\begin{matrix}{{F_{1\omega}(z)} = {\frac{\partial{C(z)}}{\partial z}\left( {V_{tip} - V_{s}} \right)V_{ac}}} & \text{Equation~~10}\end{matrix}$where z is the tip-surface separation, and C(z) is the tip-surfacecapacitance.

The local phase shift between the function generator 20 output and thecantilever oscillation is thus φ_(c)+φ on the left of the grain boundaryand φ_(c)+φ_(gb)+φ on the right of grain boundary. Hence, the change inthe phase shift of cantilever oscillations across the grain boundarymeasured by SPM is equal to the true grain boundary frequency shiftφ_(gb). The spatially resolved phase shift signal thus constitutes theelectrostatic phase angle image of the biased device. Equations 3through 10 suggest that phase shift is independent of imaging conditions(tip bias, tip surface separation, driving bias), and therefore,provides reliable data on the local surface properties.

To establish the validity of this technique, phase and amplitude imageswere acquired under varying imaging conditions. To quantify theexperimental data, the average amplitude and phase of the tip responsewere defined as the averages of unprocessed amplitude and phase images.To analyze the phase shift, the grain boundary phase profiles wereaveraged over 16 lines, extracted and fitted by a Boltzman functionφ=φ₀+Δφ_(gb)(1+exp((x−x ₀)/w))⁻¹, where w is the width and x₀ is thecenter of phase profile.

The driving frequency dependence of the average phase shift andamplitude are found to be in agreement with Equations 7a and 7b. Theamplitude was found to be linear in tip bias, as shown in FIG. 3 a, inan excellent agreement with Equation 10. The amplitude is nullified whenthe tip bias is V_(dc)=0.28±0.02 V independent of tip-surfaceseparation. Equation 10 implies that this condition is achieved whenV_(dc)=V_(s), thus yielding the value of surface potential. The phase ofthe response changes by 180° between V_(dc)=0 and V_(dc)=1 indicative oftransition from a repulsive to an attractive interaction, as shown inFIG. 3 c. The grain boundary phase shift is found to be independent oftip bias, as shown in FIG. 3 e. A small variation in the grain boundaryphase shift occurs when tip potential, V_(dc), is close to the surfacepotential, V_(s), and the amplitude of cantilever response is small.This results in a large error in the phase signal and associated highernoise level of the phase image. Note that the slopes of lines in FIG. 3a are smaller for large tip-surface separations, indicative of adecrease in capacitive force. At the same time, the grain boundary phaseshift does not depend on distance. The amplitude is linear in drivingbias, V_(ac), as shown in FIG. 3 b. Both the average and grain boundaryphase shift are essentially driving amplitude independent, as shown inFIGS. 3 d and 3 e.

To summarize, experimental observations indicate that the amplitude ofthe cantilever response is linear in tip bias and driving voltage and isa complex function of tip-surface separation and driving frequency, inagreement with Equations 9b and 10. At the same time, grain boundaryphase shift is independent of tip surface separation, tip bias, anddriving voltage provided that the response is sufficiently strong to beabove the noise level. Hence, it can be attributed to the phase shift ofac bias through the grain boundary. Further, tip bias, frequency anddriving amplitude dependence of cantilever response to sample ac biaswere found to be in excellent agreement with the theoretical model.

The frequency dependencies of average and grain boundary phase shift areshown in FIGS. 4 a and 4 b. The average phase shift changes from 0 forω<<ω_(r) to 180° for ω>>ω_(r) is in agreement with Equation 9b, whilethe grain boundary phase shift decreases with driving frequency. Aspredicted by Equation 3, the product ω tan(φ_(gb)) is frequencyindependent and is equal to RC_(gb). Calculated values of ω tan(φ_(gb))as a function of ω are shown in FIG. 4 b. The product is substantiallyconstant and from the experimental data, R_(gb)C_(gb)≈1.1·10⁻³ s.Analysis in the vicinity of the resonant frequency of the cantilever iscomplex due to the cross talk between ac driving and cantileveroscillations during topography acquisition and the resonant frequencyshift of the cantilever due to electrostatic force gradients. Forfrequencies above ω_(r) the amplitude of the response decreases rapidlywith frequency, as indicated by Equation 9a, therefore, the phase errorincreases. Imaging is possible for frequencies as small as ω_(r)/2.According to Equations 9a and 9b, response at this frequency isessentially equal to the response at zero frequency (dc limit for weakdamping). This implies that the frequency range is limited by theacquisition time of lock-in amplifier 27 and scan rate rather thancantilever sensitivity. Consequently, there is no fundamental limitationon imaging at the low frequencies; moreover, spectroscopic variants ofthis technique can in principle be performed in all frequency rangesbelow the cantilever resonant frequency ω_(r). This yields the localphase angle shift at the grain boundary and determines the product ofgrain boundary resistivity and capacitance.

To determine the individual values for grain boundary resistivity andcapacitance, several techniques may be employed including SSPM,Electrostatic Force Microscopy (EFM), Scanning Tunneling Potentiometry(STP), and the like. SSPM was used to determine grain boundaryresistivity for the bicyrstal described above; however, other techniquesmay be used to determine resistivity. An equivalent circuit for a dcbiased grain boundary is shown in FIG. 5 a. The resistors, R, correspondto the known current limiting resistors in the experimental setup, whileR₀ and R_(v) correspond to the ohmic (e.g., due to the imperfectionsand/or surface conductivity) and non-ohmic (varistor) components ofgrain boundary conductivity, respectively. The conductivity of the grainboundary region is:σ=σ₀ +γV _(v) ^(α),  Equation 11where σ is the conductivity of the bicrystal, σ₀=1/R₀, α is the varistornonlinearity coefficient and γ is the corresponding prefactor. Thepotential drop at the grain boundary, V_(v), is related to the total dcbias, V, applied to the circuit as:V=(1+2Rσ ₀)V _(v)+2RγV _(v) ^(α)  Equation 12

Sample bias dependence of grain boundary potential drop is compared toEquation 12. The ohmic contribution to the grain boundary conductivityis σ₀=(3.8±0.6)·10⁻³ Ohm⁻¹. The nonlinearity coefficient α≈3.5 isrelatively low and within the expected range for SrTiO₃ based varistors.In comparison, two-probe I-V measurements between copper contact padsyield α=2.7±0.01 and σ₀=(1.59±0.01)·10⁻³ Ohm⁻¹.

Having calculated the ohmic contribution to grain boundary conductivity,the capacitive contribution can be calculated as 1.3×10⁻¹ F/m². Usingthe value of interface resistance from the I-V measurements yields amore accurate estimate of grain boundary capacitance as 5.5×10⁻² F/m².The values are in agreement with results determined from two probetransport measurements taken at 20 kHz (C_(gb)=5.4×10⁻² F/m²) and fourpole transport measurements (C_(gb)=5.9×10⁻² F/m²).

Experimental data was also acquired for a metal-semiconductor interface.SIM phase profiles across the metal-semiconductor interface as afunction of bias for different circuit terminations are shown in FIGS. 7a and 7 b. FIGS. 7 a and 7 b were produced using the techniquesdescribed above and show a phase profile across the interface fordifferent lateral biases for R=500 Ohm for FIGS. 7 a and 100 kOhm forFIG. 7 b.

For a small current limiting resistor, the phase shift across theinterface is anomalously large, ˜172° at 3 kHz and 106° at 100 kHz.Phase shifts φ_(d)>90° imply that application of negative bias to thedevice results in the increase of surface potential. This behavior issimilar to dc potential behavior observed in SSPM measurements and isattributed to photoelectric carrier generation in the junction region.Again, this effect is completely suppressed by current limitingresistors R=10 kOhm and larger and phase shift at the interface for 100kOhm termination is shown in FIG. 7 b. Note that for a forward biaseddevice, phase shift on the left is voltage independent, while there issome residual phase shift on the right of the Schottky barrier. Thisphase shift is attributable to the diffusion capacitance of a forwardbiased junction.

Frequency dependence of tip oscillation phase and amplitude on the leftand on the right of the junction is shown in FIGS. 8 a and 8 b. FIGS. 8a and 8 b show frequency dependence of tip oscillation phase (FIG. 8 a)and amplitude (FIG. 8 b) on the left and on the right of the junctionfor R=10 kOhm (▾, ▴) and 100 kOhm (▪,●). It can be seen that theresulting tip dynamics is rather complex and is determined by theconvolution of the harmonic response of tip oscillations to periodicbias and frequency dependence of voltage oscillation phase and amplitudeinduced by local bias. Nevertheless, the abrupt phase change of about180° and tip oscillation amplitude maximum at f=72 kHz are indicative ofmechanical tip resonance. Detailed analysis of frequency-dependence ofthe amplitude has demonstrated that resonant frequency on the left andright of the junction are shifted by ˜1 khz due to the difference insurface potential and electrostatic force gradient. As suggested byEquations 9a and 9b, phase and amplitude characteristics of a harmonicoscillator are very steep close to the resonant frequency of theoscillator. Therefore, minute changes of the resonance frequency resultsin major errors for phase and amplitude data in this frequency region.To minimize this effect, the data used for quantitative analysis offrequency dependence of phase and amplitude signal was collected from 3kHz to 65 kHz and 75 kHz to 100 kHz.

Frequency dependence of phase shift for different circuit currentlimiting resistors is shown in FIG. 9 a. FIGS. 9 a and 9 b showfrequency dependence of tan(φ_(d)) and amplitude ratios, respectively,for circuit terminations of 10 kOhm (▪), 47 kOhm (●), 100 kOhm (▾) and220 kOhm (▴). In FIG. 9 a, solid lines are linear fits and fittingparameters are summarized in Table I. In FIG. 9 b, solid lines arecalculated amplitude ratios. From impedance spectroscopy data therelaxation frequency of the junction is estimated as 1.5 kHz at −5Vreverse bias. Therefore, measurements are performed in the highfrequency region in which Equation 4 is valid. In agreement withEquation 4, tan(φ_(d)) is inversely proportional to frequency with aproportionality coefficient determined by the product of interfacecapacitance and resistance of the current limiting resistor (subscriptsof d correspond to the interface of the metal semiconductor device). Inthe vicinity of the resonant frequency of the cantilever (f₀=72 kHz ),difference in the force gradient acting on the probe on the left andright of the Schottky barrier results in the shift of cantileverresonant frequency and erratic phase shifts and this region was excludedfrom data analysis. The experimental data were approximated by functionlog(tan(φ_(d)))=a+b log(f) and corresponding fitting parameters arelisted in Table I.

TABLE I Frequency dependence of phase shift R, kOhm a b C_(d), 10⁻¹⁰ FV_(d), V 10 4.94 ± 0.02 −0.99 ± 0.01 1.83 4.83 47 4.21 ± 0.01 −0.98 ±0.01 2.11 3.85 100 3.84 ± 0.01 −0.98 ± 0.01 2.32 2.86 220 3.29 ± 0.04−0.98 ± 0.02 3.76 0.80Note that coefficient b is within experimental error from theoreticalvalue b=−1, in agreement with selected parallel R-C model for theinterface. As follows from Equation 4, interface capacitance can bedetermined as C_(d)=10^(−α)/(2πR) and capacitance values for differentcircuit terminations are listed in Table I. Note that interfacecapacitance increases with the value of current limiting resistor and inall cases is larger than capacitance obtained from impedancespectroscopy, C_(d)=1.71 10⁻¹⁰ F. Amplitude ratio was calculated andcompared with experimental results with voltage correction illustratedin FIG. 9(b). Note the agreement between experimental and calculatedvalues despite the absence of free parameters. In general, one wouldexpect worse agreement between theory and experiment for tip oscillationamplitude data due to the high sensitivity of electrostatic forces, and,therefore, oscillation amplitude, to the variations of tip-surfacecapacitance due to surface roughness.

The increase of capacitance for large R is due to the well-knowndependence of junction capacitance on the bias. As can be seen in FIG.6, potential drop at the interface at given lateral bias is smaller forlarge R. FIG. 6 is a graph showing potential drop at the interface as afunction of lateral bias for current limiting resistors of 10 kOhm(solid line), 47 kOhm (long dashed line), 100 kOhm (short dashed line),220 kOhm (alternating dash-dot line), and 1 MOhm (dash-dot-dot line).The slopes and intercepts of the lines in the reverse bias regime aresummarized in Table. I. A small potential drop corresponds to smalldepletion width and high junction capacitance. To verify thisassumption, the phase shift at the interface was measured as a functionof lateral dc bias for different circuit terminations and thecorresponding dependence is shown in FIG. 10 a. FIG. 10 a shows voltagephase angle tangent tan(φ_(d)) as a function of lateral bias for currentlimiting resistors of 10 kOhm (▪), 47 kOhm (●), 100 kOhm (▾) and 220kOhm (▴). FIG. 10 b shows 1/C² vs. V_(d) for different current limitingresistors. Note that although tan(φ_(d)) varies by 2 orders of magnitudefrom R=10 kOhm to R=220kOhm, 1/C² exhibits universal behavior. Also notethat under reverse bias (V=−5 V) conditions tan(φ_(d)) changes by almosttwo orders of magnitude from tan(φ_(d))=1.8 for R=10 kOhm totan(φ_(d))=0.042 for R=220 kOhm. Interface capacitance vs. lateral biasdependence can be calculated from the data in FIG. 10 a. At the sametime, potential drop at the interface vs. lateral bias is directlyaccessible from the SSPM measurements. Combination of the two techniquesallows one to determine the C-V characteristic of the interface.Calculated 1/C² vs. V_(d) dependence is shown in FIG. 10 b. Note thatthe resulting curve shows universal behavior independent of the value ofthe current limiting resistor. This dependence is approximated by:

 1/C ²=(4.0±0.6)·10¹⁸+(6.1±0.3)·10¹⁸ V _(d)  Equation 13

For an ideal metal-semiconductor junction, voltage dependence ofcapacitance is given by: $\begin{matrix}{\frac{1}{C^{2}} = {\frac{2}{q\quad ɛ_{s}N_{B}}\left( {\phi_{B} - V - \frac{2{kT}}{q}} \right)}} & \text{Equation~~14}\end{matrix}$

where ε_(s)=11.9, ε₀ is dielectric permittivity for silicon, and N_(B)is dopant concentration. Comparison of experimental data and Equations13 and 14 allows estimation of Schottky barrier height as φ_(B)=0.6±0.1V, which is in agreement with the Schottky barrier height obtained fromI-V measurements (φ_(B)=0.55 V). From the slope of the line the dopantconcentration for the material is estimated as N_(B)=1.06 10²⁴ m⁻³.

Therefore, local interface imaging of the metal-semiconductor interfaceallows junction properties to be obtained that are consistent withproperties determined by macroscopic techniques.

In addition to determining grain boundary resistance and capacitancevalues for a sample having an interface, magnetic force data can bedetermined for a current carrying sample. Particularly, magnetic forcedata that is not magnetic gradient force data can be determined.Further, errors from surface potential electrostatic interactions may bereduced by canceling the sample surface potential. This technique isreferred to herein as potential correction magnetic-force microscopy(PMFM) and utilizes a magnetic tip 15. Tip 15 may be coated with amagnetic coating or may be constructed entirely or partially of magneticmaterial. The first harmonic of the force acting on the dc-biased tip,while an ac bias is applied to the sample, can be described as:$\begin{matrix}{{F_{1\omega}(z)} = {{\frac{\partial{C(z)}}{\partial z}\left( {V_{dc}^{tip} - V_{surf}} \right)V_{ac}^{loc}\quad{\sin\left( {\omega\quad t} \right)}} + {q\quad B_{ac}\sin\quad\left( {\omega\quad t} \right)}}} & \text{Equation~~15}\end{matrix}$The first term in Equation 15 originates from capacitive tip to surfaceinteractions and is linear with respect to dc tip bias, V^(tip) _(dc).The second term originates from magnetic forces and it is assumed thatthe magnetic interaction is limited only to the first order magneticmonopole, q. The magnetic field amplitude B_(ac) is proportional to thecurrent I_(ac) in the circuit and can be calculated from the Biot-Savartlaw. The current is proportional to the driving voltage V_(ac), appliedto the line. The relationship between the force, magnitude, and phase ofthe response of a damped harmonic oscillator with resonant frequency ω₀to the periodic force F(t)=F_(1ω) cos(ωt) are given by Equations 9a and9b above.

A first approach, referred to herein as basic PMFM mode, reduces theimpact of the electrostatic interactions by adjusting the dc tip bias toa constant dc voltage to offset the electrostatic interaction betweenthe tip and the surface of the sample. FIG. 11 is an optical micrographof a sample to which this approach was applied. As shown in FIG. 11,area 1 is a single line area and area 2 is a double line area. FIGS. 12a through 12 f shows graphical results of data taken from area 1 of thesample of FIG. 11 using the system of FIG. 1 and basic MFM mode. FIG. 12a illustrates a calculated force profile for area 1 of the sample andFIG. 12 b illustrates a measured force profile using basic PMFM mode fora tip dc bias of −2 V. FIG. 12 c illustrates a calculated force profilefor area 1 of the sample and FIG. 12 b illustrates a measured forceprofile using basic PMFM mode for a tip dc bias of 316 mV. FIG. 12 eillustrates a calculated force profile for area 1 of the sample and FIG.12 f illustrates a measured force profile using basic PMFM mode for atip dc bias of 2 V. The arrows indicate topographic artifacts that werenot reproducible on the trace and retrace paths. It should be noted thatthe graphs of FIGS. 12 a, 12 b, 12 e, and 12 f each include bothmagnetic and electrostatic contributions. In the graphs of FIGS. 6 c and6 d, however, electrostatic contributions are significantly reducedbecause they have been offset by the dc tip bias of 316 mV. This bias isreferred to as the nulling bias because it corresponds to a theoreticalminimal amount of electrostatic interaction between the tip and thesurface.

The magnetic and electrostatic components for basic PMFM mode areillustrated in FIGS. 13 a through 13 d. FIG. 13 a is a graph of forceversus distance that illustrates a force profile in basic PMFM modewhere φ₁ represents the magnetic contribution to the force, φ₂represents the capacitive interaction between the tip and the line, andφ₃ represents the capacitive interaction between the tip and the sample.FIG. 13 b is a graph for tip bias versus measured force that representsthe tip bias dependence of φ₁, φ₂, and φ₃. FIG. 13 c is a graph of linebias versus measured force at the nulling bias. FIG. 13 d is a graph offrequency versus measured force. The solid line was fitted according toEquations 9a and 9b.

Adjusting the tip bias, however, provides an average correction forcapacitive contributions over the length of the sample. This canintroduce errors to measurements of samples having non-uniformpotentials over the length of the sample. To overcome this, localpotentials can be determined with voltage modulation techniques, such asfor example, scanning surface potential microscopy (SSPM). In SSPM, thetip is biased with a signal that can have both an ac and a dc component,as described by V_(tip)=V_(dc)+V_(ac) cos(ω_(r)t), where ω_(r) isselected to be the resonant frequency of the cantilever to provide astrong mechanical response. The first harmonic of the force on thecantilever due to capacitance is given by Equation 12. $\begin{matrix}{{F_{1\omega}^{cap}(z)} = {\frac{\partial{C(z)}}{\partial z}\left( {V_{dc}^{tip} - V_{surf}} \right)V_{ac}^{tip}{\sin\left( {\omega\quad t} \right)}}} & \text{Equation~~16}\end{matrix}$

Using this relationship V_(surf) is determined and feedback control isused to set the dc component of the tip to V_(surf), thereby mapping andcanceling the surface potential simultaneously. It is noteworthy thatEquations 15 and 16 are similar, the difference being the secondmagnetic term of Equation 15.

FIGS. 14 a through 14 d illustrate experimental results using the abovedescribed technique. FIG. 14 a is a graph of lateral sample distance forarea 1 of the sample versus determined force with the tip biased at 316mV. FIG. 14 b is a graph of lateral sample distance for area 1 of thesample versus determined potential using SSPM with a line frequency of70 kHz, a tip frequency (at the resonant frequency ω_(r)) of 75.1679kHz, and V_(ac) of 5 V. FIG. 14 c is a graph of lateral sample distancedetermined from the difference of the force and potential profiles ofFIGS. 14 a, 14 b, respectively, using a phenomenological proportionalitycoefficient. As can be seen in FIG. 14 c, topographical artifacts can bereduced. FIG. 14 d is a graph of lateral sample distance versus force ofexperimental data measured using an ac bias applied to the tip at theresonant frequency of the cantilever while conventional SSPM techniquesand feedback are used to match the dc bias of the tip to the determinedsurface potential.

PMFM may be implemented with three passes: a first pass measures thesurface topography of the sample, a second pass measures the surfacepotential of the sample, and the third pass determines magnetic forcedata as a conventional MFM while the dc bias of the tip is adjustedbased on the values determined in the second pass. Alternatively, PMFMmay be implemented in one pass, wherein for each pixel along the pathtaken by the tip, the sample surface topography is measured, the surfacepotential is determined, and magnetic force data is determined with thedc bias of the tip adjusted based on the determined surface potential.Two pass implementations are also contemplated.

Portions of the invention may be embodied in the form of program code(i.e., instructions) stored on a computer-readable medium, such as amagnetic, electrical, or optical storage medium, including withoutlimitation a floppy diskette, CD-ROM, CD-RW, DVD-ROM, DVD-RAM, magnetictape, flash memory, hard disk drive, or any other machine-readablestorage medium, wherein, when the program code is loaded into andexecuted by a machine, such as a computer, the machine becomes anapparatus for practicing the invention. When implemented on ageneral-purpose processor, the program code combines with the processorto provide a unique apparatus that operates analogously to specificlogic circuits.

It is noted that the foregoing illustrations have been provided merelyfor the purpose of explanation and are in no way to be construed aslimiting of the invention. While the invention has been described withreference to illustrative embodiments, it is understood that the wordswhich have been used herein are words of description and illustration,rather than words of limitation. Further, although the invention hasbeen described herein with reference to particular structure, methods,and embodiments, the invention is not intended to be limited to theparticulars disclosed herein; rather, the invention extends to allstructures, methods and uses that are within the scope of the appendedclaims. Those skilled in the art, having the benefit of the teachings ofthis specification, may effect numerous modifications thereto andchanges may be made without departing from the scope and spirit of theinvention, as defined by the appended claims.

1. A method for determining impedance information of an interface in asample, the method comprising the steps of: (a) applying an ac voltageto the sample, laterally across the interface, the ac voltage having apredetermined frequency; (b) disposing a cantilevered tip in a firstposition proximate to a surface of the sample; (c) measuring a firstresponse of the cantilevered tip with the cantilevered tip in the firstposition; (d) placing the cantilevered tip in a second positionproximate to the surface of the sample, the interface being between thefirst position and the second position; (e) measuring a second responseof the cantilevered tip with the cantilevered tip in the secondposition; and (f) determining impedance information of the interfacebased upon the measured first response and the measured second response.2. The method as recited in claim 1, wherein the step of: measuring afirst response comprises measuring a first phase angle of deflection ofthe cantilevered tip; measuring a second response comprises measuring asecond phase angle of deflection of the cantilevered tip; anddetermining impedance information comprises: determining a phase shiftbased upon the first phase angle and the second phase angle; anddetermining impedance information of the interface based upon the phaseshift and the frequency of the ac voltage.
 3. The method as recited inclaim 2, wherein the step of determining impedance information furthercomprises determining an impedance product of the interface accordingto:${\tan\quad\left( \varphi_{gb} \right)} = \frac{\omega\quad C_{gb}R_{gb}^{2}}{\left( {R + R_{gb}} \right) + {R\quad\omega^{2}C_{gb}^{2}R_{gb}^{2}}}$where C_(gb) is the capacitance of the interface; R_(gb) is theresistance of the interface; ω is the frequency of the ac voltage;φ_(gb) is the phase shift; and R is a resistance of a current limitingresistor in series with the sample.
 4. The method as recited in claim 2,further comprising the step of: selecting the frequency ω of the acvoltage such that tan(ω_(gb)) is proportional to ω⁻¹; and wherein thestep of determining impedance information further comprises determiningthe capacitance of the interface according to:${\tan\quad\left( \varphi_{gb} \right)} = \frac{1}{\omega\quad R\quad C_{gb}}$where C_(gb) is the capacitance of the interface; ω is the frequency ofthe ac voltage; φ_(gb) is the phase shift; and R is a resistance of acurrent limiting resistor in series with the sample.
 5. The method asrecited in claim 2, further comprising the step of: selecting thefrequency ω of the ac voltage such that tan(φ_(gb)) proportional to ω;and wherein the step of determining impedance information furthercomprises determining the resistance of the interface according to:${\tan\quad\left( \varphi_{gb} \right)} = \frac{\omega\quad C_{gb}R_{gb}^{2}}{\left( {R + R_{gb}} \right)}$where R_(gb) is the resistance of the interface; C_(gb) is thecapacitance of the interface; ω is the frequency of the ac voltage;φ_(gh) is the phase shift; and R is a resistance of a current limitingresistor in series with the sample.
 6. The method as recited in claim 1,wherein the step of: measuring a first response comprises measuring afirst amplitude of deflection of the cantilevered tip; measuring asecond response comprises measuring a second amplitude of deflection ofthe cantilevered tip; and determining impedance information comprises:determining an amplitude ratio based upon the measured first amplitudeand the measured second amplitude; and determining an impedance of theinterface based upon the amplitude ratio and the frequency of the acvoltage.
 7. The method of claim 6, wherein the step of determining theimpedance further comprises determining the impedance according to:$\beta^{- 2} = \frac{\left\{ {\left( {R + R_{gb}} \right) + {R\quad\omega^{2}C_{gb}^{2}R_{gb}^{2}}} \right\}^{2} + {\omega^{2}C_{gb}^{2}R_{gb}^{4}}}{{R^{2}\left( {1 + {\omega^{2}C_{gb}^{2}R_{gb}^{2}}} \right)}^{2}}$where β is amplitude ratio across the interface, C_(gb) is thecapacitance of the interface, R_(gb) is the resistance of the interface,ω is the frequency of the ac voltage, and R is a resistance of a currentlimiting resistor in series with the sample.
 8. The method as recited inclaim 1, wherein the step of: measuring a first response comprises thesteps of: measuring a first phase angle of deflection of thecantilevered tip; and measuring a first amplitude of deflection of thecantilevered tip; measuring a second response comprises: measuring asecond phase angle of deflection of the cantilevered tip; and measuringa second amplitude of deflection of the cantilevered tip; anddetermining impedance information comprises: determining a phase shiftbased upon the measured first phase angle and the measured second phaseangle; determining an amplitude ratio based upon the measured firstamplitude and the measured second amplitude; and determining animpedance of the interface based upon the phase shift and the amplituderatio.
 9. The method as recited in claim 8, wherein the step ofdetermining an impedance of the interface comprises solving thefollowing equations:$\beta^{- 2} = \frac{\left\{ {\left( {R + R_{gb}} \right) + {R\quad\omega^{2}C_{gb}^{2}R_{gb}^{2}}} \right\}^{2} + {\omega^{2}C_{gb}^{2}R_{gb}^{4}}}{{R^{2}\left( {1 + {\omega^{2}C_{gb}^{2}R_{gb}^{2}}} \right)}^{2}}$${\tan\quad\left( \varphi_{gb} \right)} = \frac{\omega\quad C_{gb}R_{gb}^{2}}{\left( {R + R_{gb}} \right) + {R\quad\omega^{2}C_{gb}^{2}R_{gb}^{2}}}$where β is amplitude ratio across the interface, C_(gb) is thecapacitance of the interface, R_(gb) is the resistance of the interface,φ_(gb) is the phase shift; ω is the frequency of the ac voltage, and Ris a resistance of a current limiting resistor in series with thesample.
 10. The method as recited in claim 8, further comprising thesteps of: repeating each step of claim 8, for each of a predefinedplurality of ac voltage frequencies; and determining an impedance of theinterface as a function of frequency based upon the determined phaseshifts and amplitude ratios.
 11. The method as recited in claim 8wherein the step of determining an impedance further comprisesdetermining the impedance of the interface based upon the phase shiftand amplitude ratio and the frequency of the ac voltage by using thecircuit termination comprised of average resistive and capacitivecomponents determined by conventional impedance spectroscopy bymeasuring the frequency dependence of phase angle or measuring phaseangle and amplitude ratio at a single frequency with a least square fitprocedure.
 12. The method as recited claim 8 further comprising thesteps of: repeating each step of claim 8 and applying a first dc signallaterally across the interface while performing each measuring step;repeating each step of claim 8 and applying a second dc signal laterallyacross the interface while performing each measuring step; anddetermining dc signal bias dependence of the interface resistance andcapacitance based upon the first and second dc signal.